Integrable tops and non-commutative torus (original) (raw)
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Physica D: Nonlinear Phenomena, 1993
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Poisson–Lie groups, bi-Hamiltonian systems and integrable deformations
Journal of Physics A: Mathematical and Theoretical
Given a Lie-Poisson completely integrable bi-Hamiltonian system on R n , we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group Gη of dimension n, where η ∈ R is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on Gη that underly the dynamics of the deformed system and by making use of the group law on Gη, one may obtain two completely integrable Hamiltonian systems on Gη × Gη. By construction, both systems admit reduction, via the multiplication in Gη, to the deformed bi-Hamiltonian system in Gη. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.
Poisson-Lie group, bi-Hamiltonian system and integrable deformations
2016
Given a Lie-Poisson completely integrable bi-Hamiltonian system on R n , we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group Gη of dimension n, where η ∈ R is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on Gη that underly the dynamics of the deformed system and by making use of the group law on Gη, one may obtain two completely integrable Hamiltonian systems on Gη × Gη. By construction, both systems admit reduction, via the multiplication in Gη, to the deformed bi-Hamiltonian system in Gη. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.
Poisson maps and integrable deformations of the Kowalevski top
Journal of Physics A: Mathematical and General, 2003
We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the Lie algebras e(3) and so(4). Using this map we establish a connection between the deformed Kowalevski top on e(3) proposed by Sokolov and the Kowalevski top on so(4). The connection between these systems leads to the separation of variables for the deformed system on e(3) and yields the natural 5 × 5 Lax pair for the Kowalevski top on so(4).
Deformation of surfaces, integrable systems and Self-Dual Yang-Mills equation
2002
A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern-Simons action via reduction of the gauge connection in Hermitian symmetric spaces. In this paper we show that the methods developed in studying classical non-Abelian pure Chern-Simons actions, can be naturally implemented by means of a geometrical interpretation of such systems. The Chern-Simons equation of motion turns out to be related to time evolving 2-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss-Mainardi-Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.
Integrable GL(2) geometry and hydrodynamic partial differential equations
Communications in Analysis and Geometry, 2010
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Acta Applicandae Mathematicae, 2008
Using Grozman's formalism of invariant differential operators we demonstrate the derivation of N = 2 Camassa-Holm equation from the action of Vect(S 1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N = 2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H 1-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N = 1 super peakon type equations, known as N = 1 super b-field equations, are derived from the action of Vect(S 1|1) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S 1|1) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N = 1 super b-field equations. Keywords Pseudo-differential symbols • Super KdV • Camassa-Holm equation • Geodesic flow • Super b-field equations • Moyal deformation • Noncommutative integrable systems Mathematics Subject Classification (2000) 17B68 • 37K10 • 58J40 1 Prelude to Noncommutative Integrable Systems Noncommutative geometry [5] extends the notions of classical differential geometry from differential manifold to discrete spaces, like finite sets and fractals, and noncommutative spaces which are given by noncommutative associative algebras. It was an idea of Descartes that we can study a space by means of functions on the space, in other words, the algebra
Poisson Quasi-Nijenhuis Manifolds and the Toda System
Mathematical Physics, Analysis and Geometry, 2020
The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by an example with 3 degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we prove that the closed (or periodic) n-particle Toda lattice can be framed in such a geometrical structure, and its well-known integrals of the motion can be obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that is, a tensor field N of type (1, 1) defined on the phase space of the lattice. This example and some of its generalizations are used to understand whether one can define in a reasonable sense a notion of involutive Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non periodic) and the closed Toda systems is also framed in the context of a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds.